Biorthogonal system

About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.

Source Localization of EEG Brainwaves Activities via Mother Wavelets Families for SWT Decomposition

Abstract: A Brain-Computer Interface (BCI) is a system used to communicate with an external world through the brain activity. The brain activity is measured by electroencephalography (EEG) signal and then processed by a BCI system. EEG source reconstruction could be a way to improve the accuracy of EEG classification in EEG based brain-computer interface (BCI). The source localization of the human brain activities can be an important resource for the recognition of the cognitive state, medical disorders, and a better understanding of the brain in general. In this study, we have compared 51 mother wavelets taken from 7 different wavelet families, which are applied to a Stationary Wavelet Transform (SWT) decomposition of an EEG signal. This process includes Haar, Symlets, Daubechies, Coiflets, Discrete Meyer, Biorthogonal, and reverse Biorthogonal wavelet families in extracting five different brainwave subbands for source localization. For this process, we used the Independent Component Analysis (ICA) for feature extraction followed by the Boundary Element Model (BEM) and the Equivalent Current Dipole (ECD) for the forward and inverse problem solutions. The evaluation results in investigating the optimal mother wavelet for source localization eventually identified the sym20 mother wavelet as the best choice followed by bior6.8 and coif5.

Non-Hermitian topology in monitored quantum circuits

Abstract: We demonstrate that genuinely non-Hermitian topological phases and corresponding topological phase transitions can be naturally realized in monitored quantum circuits, exemplified by the paradigmatic non-Hermitian Su-Schrieffer-Heeger model. We emulate this model by a 1D chain of spinless electrons evolving under unitary dynamics and subject to periodic measurements that are stochastically invoked. The non-Hermitian topology is visible in topological invariants adapted to the context of monitored circuits. For instance, the topological phase diagram of the monitored realization of the non-Hermitian Su-Schrieffer-Heeger model is obtained from the biorthogonal polarization computed from an effective Hamiltonian of the monitored system. Importantly, our monitored circuit realization allows direct access to steady state biorthogonal expectation values of generic observables, and hence, to measure physical properties of a genuine non-Hermitian model. We expect our results to be applicable more generally to a wide range of models that host non-Hermitian topological phases.

Existence and design of biorthogonal matrix-valued wavelets

Abstract: In this paper, we study biorthogonal matrix-valued wavelets for analyzing matrix-valued signals based on matrix multiresolution analysis. Firstly, sufficient conditions for the existence of biorthogonal matrix-valued scaling function are established in terms of two-scale matrix symbols. Then we focus on the construction of biorthogonal matrix-valued wavelet. Two designs based on factorizations of biorthogonal two-scale matrix symbol are presented. In particular, explicit constructing formulations for biorthogonal matrix-valued wavelets are given. With these formulations, highpass filters {Gk} and {Gk} of biorthogonal matrix-valued wavelets can be given explicitly by lowpass filters {Hk} and {Hk} of their corresponding biorthogonal scaling functions. Finally, according to our designs, two examples of two-scale matrix filters are given.

Biorthogonal wavelets on the spectrum

Abstract: In this article, we introduce the notion of biorthgonoal nonuniform multiresolution analysis on the spectrum $\Lambda=\left\{0, r/N\right\}+2\mathbb Z$, where $N\ge 1$ is an integer and $r$ is an odd integer with $1\le r\le 2N-1$ such that $r$ and $N$ are relatively prime. We first establish the necessary and sufficient conditions for the translates of a single function to form the Riesz bases for their closed linear span. We provide the complete characterization for the biorthogonality of the translates of scaling functions of two nonuniform multiresolution analysis and the associated biorthogonal wavelet families. Furthermore, under the mild assumptions on the scaling functions and the corresponding wavelets associated with nonuniform multiresolution analysis, we show that the wavelets can generate Reisz bases.

Biorthogonal wavelets for subdivision volumes

Abstract: We present a biorthogonal wavelet construction based on Catmull-Clark-style subdivision volumes. Our wavelet transform is the three-dimensional extension of a previously developed construction of subdivision-surface wavelets that was used for multiresolution modeling of large-scale isosurfaces. Subdivision surfaces provide a flexible modeling tool for surfaces of arbitrary topology and for functions defined thereon. Wavelet representations add the ability to compactly represent large-scale geometries at multiple levels of detail. Our wavelet construction based on subdivision volumes extends these concepts to trivariate geometries, such as time-varying surfaces, free-form deformations, and solid models with non-uniform material properties. The domains of the repre-sented trivariate functions are defined by lattices composed of arbitrary polyhedral cells. These are recursively subdivided based on stationary rules converging to piecewise smooth limit-geometries. Sharp features and boundaries, defined by specific polygons, edges, and vertices of a lattice are explicitly represented using modified subdivision rules. Our wavelet transform provides the ability to reverse the subdivision process after a lattice has been re-shaped at a very fine level of detail, for example using an automatic fitting method. During this coarsening process all geometric detail is compactly stored in form of wavelet coefficients from which it can be reconstructed without loss.